The Site Frequency Spectrum in an Exponentially-Growing Population with Selection

We consider a supercritical two-type continuous-time linear birth-death process with mutation and selection, in which wild-type individuals give rise to mutant offspring with a larger net growth rate. In this setting, we investigate the ``driver'' site frequency spectrum (SFS), or the random measure that records mutant allelic frequencies in the population. We examine various regions of the SFS. First, we derive exact moments and prove results concerning the mean behavior of the driver SFS at large times and frequencies. Strong laws of large numbers for the driver SFS are proven by constructing suitable $L^2$-approximations. These results are extended to the setting in which the fitness increase associated with each clone is random. Next, we allow the frequencies to vary with time to examine the number of ``intermediate'' and ``large'' clones. Using this, we find a cutoff frequency at which there are order 1 number of clones. Our results allow for estimation of relevant evolutionary parameters, such as the fitness increase of mutant versus wild-type cells.